On the Mechanism of Soot Nucleation. IV. Molecular Growth of the Flattened E-Bridge

Rotationally excited dimerization of aromatic moieties, a mechanism proposed recently to explain the initial steps of soot particle inception in combustion and pyrolysis of hydrocarbons, produces a molecular structure, termed E-bridge, combining the two aromatics via five-membered aromatic rings sharing a common bond. The present study investigates a hydrogen-mediated addition of acetylene to the fused five-membered ring part of the E-bridge forming a seven-membered ring. The carried out quantum-mechanical and rate theoretical calculations indicate the plausibility of such capping reactions, and kinetic Monte Carlo simulations demonstrate their frequent occurrence. The capping frequency, however, is limited by “splitting” the fused five-membered bridge due to five-membered ring migration. A similar migration of edge seven-membered rings is shown to be also rapid but short, as their encounter with five-membered rings converts them both into six-membered rings.


INTRODUCTION
Recent proposals on combating climate change include utilization of hydrogen. 1 Technological strategies aimed at production of hydrogen in required amounts include thermal 2 or plasma-assisted 3 pyrolysis of natural gas (or, more generally, biomass) that decomposes a hydrocarbon into hydrogen and solid carbon, with the latter then used for production of, say, construction materials. This vision brings to the center the formation of carbonaceous particles in high-temperature environments that has thus far been primarily associated with their negative impact on human health 4 and the environment 5 and, hence, has been extensively studied by the combustion community.
Scientifically, the formation of carbonaceous particles in a flame is a fascinating phenomenon: the molecular transition from a few-atom hydrocarbon to a solid material composed of thousands of carbon atoms involves multiple physical and chemical processes, all occurring within a thin layer of a flame on a time scale of a millisecond, with the structure of the forming solid particles exhibiting patterns of self-organization. 6 Years of research, spanning multiple disciplines and scientific communities, provided a general understanding of the phenomena involved. 7 Yet, some questions, especially in the area of elementary chemical reactions, still remain. One of them is the specific pathway responsible for particle inception, 7−9 i.e., the transformation of polycyclic aromatic hydrocarbons (PAH)s, the established precursors, 6,10 into a "solid" phase.
Examination of various proposals made toward the particle inception mechanism 9 led us to focus on one of the most promising of them, the pathway proceeding through rotationally excited dimerization of the two colliding PAH forming Ebridge, a structure composed of two aromatic rings sharing a common bond (see Figure 1). The initially forming E-bridge, in step I of Figure 1, has the two precursor PAHs at an angle. The immediately following H-abstraction transforms the angled structure into a flattened one, 11 shown as step II in Figure 1.
The question we pose now is what happens next. Will the flattened E-bridge structure grow just at the outer aromatic structures, i.e., at the E-bridge "wings"? Or could the 5−5 bay of the flattened E-bridge be capped, thus propagating a more balanced growth? (By "capping" we imply the formation of an aromatic ring by addition of a CC moiety, such as acetylene, to a bay zone of a PAH.) With this in mind, in the present study, we investigate one of the immediate possibilities, H-mediated addition of C 2 H 2 forming a seven-membered ring, and analyze implication of this capping to the follow-up growth of the flattened E-bridge.

Quantum Chemistry.
We employed here our standard theoretical approach used earlier in the studies of the E-bridge formation, with the difference in the method for energy refinement by higher-level single-point calculations (see below). In particular, geometries of the reactants, intermediates, transition state (first-order saddle point) structures (referred to as "transition states" in the text), and products of the C 34 H 15 + C 2 H 2 and C 36 H 15 + C 2 H 2 reactions featuring two consecutive seven-membered ring closures in the bay areas of the flattened E-bridge with pyrene wings (C 34 H 16 , Figure 1) were optimized using the density functional theory (DFT) B3LYP method with the 6-311G(d,p) basis set. 12,13 Here, the C 34 H 15 radical is obtained by H abstraction from one of the bay areas of the E-bridge. In turn, the C 36 H 15 radical, which serves as the reactant in the second reaction, is obtained by H abstraction from the bay area in the C 36 H 16 product of the first reaction. Vibrational frequencies of all stationary structures were computed at the same B3LYP/6-311G(d,p) level of theory and used to evaluate zero-point vibrational energy (ZPE) corrections and for rate constant calculations. The B3LYP calculations were carried out using the GAUSSIAN 16 software package. 14 The use of B3LYP geometries and vibrational frequencies in conjunction with higher-level chemically accurate single-point energies has been repeatedly shown to provide kinetically accurate rate constants for PAH growth reactions, see e.g., refs 15 and 16. In a recent paper, 17 we have also tested the performance of B3LYP for a related ring-closure reaction vs a modern ωB97XD functional which includes dispersion corrections. It appeared that the B3LYPand ωB97XD-optimized geometries were nearly identical, and the difference in vibrational frequencies of local minima and transition states was under 4% and only ∼1% on average. Moreover, single-point G3(MP2,CC)//B3LYP and G3-(MP2,CC)//ωB97XD relative energies agreed with each other within 0.3 kcal/mol. Although the B3LYP and ωB97XD relative energies with the same basis set occasionally disagreed by up to ∼3 kcal/mol, the difference practically disappeared upon the higher-level single-point energy refinement. Here, G3(MP2,CC) calculations are not feasible, and single-point energies of the optimized structures were recalculated using domain based local pair-natural orbital singles and doubles coupled cluster method perturbatively included connected triple excitations (DLPNO-CCSD-(T)) 18,19 with Dunning's cc-pVDZ basis set. 20 In recent works, 21,22 we compared the performance of the DLPNO-CCSD(T)/cc-pVDZ method for relative energies in PAH growth (benzannulation) and E-bridge formation reactions with the chemically accurate G3(MP2,CC) approach and found a close agreement, with deviations of less than 1 kcal/ mol on average and the maximal deviation of ∼2 kcal/mol. Furthermore, the DLPNO-CCSD(T) results with the cc-pVDZ and cc-pVQZ basis sets were very close to one another with very few exceptions, where the deviation was ∼2 kcal/ mol. This comparison indicated that the chemical accuracy of about ∼2 kcal/mol can be normally achieved for molecules of this type by the DLPNO-CCSD(T) method even with the cc-pVDZ basis set, opening the opportunity to carry out accurate calculations for much larger systems, like the systems considered in the present study, than those treatable by the model chemistry G3/G4-type schemes. The DLPNO-CCSD-(T)/cc-pVDZ calculations were carried out using the ORCA quantum chemistry code. 23

Reaction Rate Coefficients.
Temperature-and pressure-dependent rate constants for the C 34 H 15 + C 2 H 2 and C 36 H 15 + C 2 H 2 reactions were evaluated using the Rice-Ramsperger-Kassel-Marcus Master Equation (RRKM-ME) approach. 24 The MESS software package 25 was utilized for the RRKM-ME calculations, where partition functions and densities of states for local mimima and numbers of states for transition states were computed within the Rigid-Rotor, Harmonic-Oscillator (RRHO) model. The Lennard-Jones parameters were estimated using the general expressions for ε and σ depending on the molecular mass proposed by Wang and Frenklach 26 for PAH molecules, whereas the parameters for the bath gas (N 2 ) were taken from the papers by Vishnyakov et al. 27,28 Our previous calculations showed that moderate changes in the Lennard-Jones parameters within the ranges corresponding to four-to six-ring PAHs result in insignificant changes of the calculated pressure-dependent rate constants of less than 10%. 29 The collisional energy transfer parameters in RRKM-ME calculations were described within the "exponential down" model, 30 where the temperature dependence of the parameter α for the deactivating wing of the energy transfer function was expressed as α(T) = α 300 (T/ 300) n , where n = 0.85 and α 300 = 247 cm −1 are "universal" values proposed by Jasper and Miller for hydrocarbons. 31 2.3. PAH Structure Evolution. PAH structure evolution was examined in kinetic Monte Carlo (kMC) simulations and burner-to-stagnation surface separation 0.8 cm). The flame was simulated with the FFCM1 34 gaseous reaction model using Cantera. 35 The stochastic evolution of PAH structure was simulated using the Gillespie algorithm. 36,37 The PAHgrowth reaction rate constants were calculated using the timedependent temperature and gaseous species profiles (H, H 2 ,

Potential Energy Surface.
Calculated potential energy diagram for the C 34 H 15 + C 2 H 2 reaction achieving a seven-membered ring closure in one of the bay areas of the flattened E-bridge with pyrene wings, following its activation through H abstraction is shown in Figure 2. At the initial step, acetylene adds to the radical site producing an initial complex i1, C 36 H 17 , overcoming a relatively low barrier of 2.3 kcal/mol at ts1. The complex is stabilized by 42.0 kcal/mol. Next, i1 can either split an H atom from the C 2 H 2 moiety producing the ethynyl-substituted E-bridge molecule p2 via ts2 or undergo a seven-membered ring closure. The H loss occurs via ts2, which resides only 2.2 kcal/mol below the initial reactants, and the product p2 is computed to be 9.4 kcal/mol exothermic relative to C 34 H 15 + C 2 H 2 . On the other hand, the ring closure process, which can take place either directly in i1 or be preceded by 1,7-H migration to form i3, is more favorable in terms of enthalpy. The i1 → i2 → p1 + H pathway, where the ring closure is followed by the H loss, leads to the bay-capped product p1, C 36 H 16 , exothermic by 36.6 kcal/mol, via transition states ts3 and ts4 positioned 23.0 and 28.1 kcal/mol lower in energy than the reactants. The second three-step pathway to p1, i1 → i3 → i4 → p1 + H, involving the 1,7-H migration, the sevenmembered ring closure, and the H elimination, is even more favorable, as the corresponding transition states ts5, ts6, and ts7 are located 28.3, 34.6, and 31.2 kcal/mol below the reactants. The intermediates i2 and i4 are connected by a 1,2-H shift via ts8, but this process is less competitive than the H losses from either of the two intermediates. It is apparent from the potential energy diagram that the formation of the thermodynamically favorable bay-capped product p1 through the well-skipping channels would compete with the production of p2, which is preferable by the entropic factor, and, depending on the pressure, with collisional stabilization of the C 36 H 17 intermediates.
Once the C 36 H 16 p1 product is formed, it can undergo the HACA sequence (i.e., H abstraction followed by acetylene addition) in the second bay area next to the E-bridge. The corresponding potential energy diagram for the C 36 H 15 + C 2 H 2 reaction is illustrated in the bottom half of Figure 2. The reaction mechanism and energetics are rather similar to those for C 34 H 15 + C 2 H 2 , and the product channels include the wellskipping pathways to the bay-capped product p1′, C 36 H 15 + C 2 H 2 → i1′ → i2′ → p1′ + H and C 36 H 15 + C 2 H 2 → i1′ → i3′ → i4′ → p1′ + H, competing with the path to the ethynylsubstituted product p2′, C 36 H 15 + C 2 H 2 → i1′ → p2′ + H. The quantitative differences in the relative energies of various intermediates and transition states are mostly minor, within the error margins of the computational method used, but with few notable exceptions. For instance, the energy of the bay-capped product p1′ with respect to the reactants increases by 5.7 kcal/ mol as compared to p1 making the reaction less exothermic. This can be attributed to the fact that C 36 H 16 p1 product is still planar, whereas C 38 H 16 p1′ loses its planarity and attains a bowl-like shape due to the accumulation of the strain energy caused by the presence of two adjacent defects in the graphenic structure consisting of fused seven-and five-membered rings. The energies of the critical transition states on the pathways to p1′, ts4′, ts5′, and ts7′, also increase by 5−6 kcal/mol as compared to ts4, ts5, and ts7 in the first bay-capping reaction C 34 H 15 + C 2 H 2 . This apparently would make the second bay capping of the E-bridge area less favorable than the capping of the first bay area. Figure 3 illustrates the calculated rate constants for the C 34 H 15 + C 2 H 2 and C 34 H 16 + C 2 H 2 reactions at 1 atm. They show a behavior similar to that for other bay-capping reactions occurring via acetylene addition to a σ PAH radical. In particular, the C 34 H 15 + C 2 H 2 is fast overall, with the total rate constant at the highpressure limit (HP) rising from 9.4 × 10 −14 to 2.3 × 10 −11 cm 3 molecule −1 s −1 in the considered 500−2500 K range. At 1 atm, a falloff of the total rate constant from the HP limit can be seen above 1200 K reaching a factor of ∼2.9 at 2500 K. At lower temperatures, the reaction is dominated by collisional stabilization of the intermediate i1 and then i4, but at T > 1250 K, the well-skipping channels leading to the bimolecular products become predominant; i4 becomes unstable above 1500 K. In the 1300−2500 K temperature range, the baycapped product p1 has the highest yield with the rate constant for its formation reaching a maximum at 2000 K. The rate constant for the formation of the ethynyl-substituted product p2 steadily grows with temperature and nearly equalizes with The Journal of Physical Chemistry A pubs.acs.org/JPCA Article that for the formation of p1 at 2500 K. At 1500 K, the baycapping rate constant involving the reaction of a sevenmembered ring closure next to the E-bridge, 3.8 × 10 −12 cm 3 molecule −1 s −1 , is similar to those for bay-capping processes with a six-membered ring formation, which were found to vary in the 8.9 × 10 −13 −7.5 × 10 −12 cm 3 molecule −1 s −1 range. 38 As mentioned above, the second seven-membered closure in the C 36 H 15 + C 2 H 2 reaction is less favorable than the closure of the first bay due to the lower exothermicity and higher barriers along the reaction path. This is indeed reflected in the calculated rate constants, displayed in Figure 3. Clearly, the collisional stabilization of i1′ and i4′ persists up to higher temperatures, p1′ is the preferred product only in the narrow temperature range around 1500 K, and the formation of p2′ prevails at 1650 K and above. The calculated bay-capping rate constant at 1500 K, 1.2 × 10 −12 cm 3 molecule −1 s −1 , is about a factor of 3 lower than that for the formation of p1 in the C 34 H 15 + C 2 H 2 reaction.

Reaction Rate Coefficients.
3.3. PAH Structure Evolution. The kMC simulations followed the evolution of the flattened E-bridge "placed" at a flame height corresponding to 1600 K, a likely location of the E-bridge formation. The simulations were carried out for a duration of 3 ms, reaching a temperature of about 1800 K. The statistics of the reaction events were collected from 1,000 kMC runs. For comparison, a similar set of calculations was performed starting with pyrene, as the E-bridge is produced from the derivatives of pyrene, in a reaction between pyrenyl and acepyrene.
Out of 1,000 runs, the first E-bridge capping, reaction C 34 H 15 + C 2 H 2 → p1 + H (Figure 2, top), occurred in 302 runs with the second capping event, reaction C 36 H 15 + C 2 H 2 → p1′ + H (Figure 2, bottom), occurring in 122 of the 302 runs; the histogram of these reaction events is displayed in the top panel of Figure 4. These results demonstrate that the capping took place in a considerable number, roughly onethird, of the cases; still, why the capping does not occur in all the cases? The analysis of the reaction pathways revealed that the reason for this is embedded ring migration.
A theoretical study of Whitesides et al. 39 suggested that fivemembered rings embedded at the graphene edges can undergo rapid H-activated migration along a zigzag edge, as illustrated in the top panel of Figure 5. Tested in kMC simulations, 37,40 this type of migration was indeed shown to be a frequent event. More recently, 32 the same mechanism of the fivemembered ring migration was shown to be one of the dominant events in the structural evolution of PAHs, exhibiting interconversion among embedded, partially embedded, and zigzag-edge five-membered rings. For a small to a moderate size PAH, such transformations have an appearance of a five-membered ring "rotation" over the perimeter of the evolving structure, as illustrated at the bottom panel of Figure  5.
The present kMC simulations revealed that the embedded five-membered ring migration is causing the split of the two "fused" five-membered rings forming the E-bridge with one of the two five-membered rings moving away from the other, as illustrated in Figure 6. This split happens to occur rather early in the simulation runs, as can be witnessed by the statistics displayed in the bottom panel of Figure 4. The early split of the "fused" five-membered ring structure destroys the E-bridge, which thus prevents the capping reaction from occurring. Still, the results show that roughly in one-third of the cases, the capping, reaction C 34 H 15 + C 2 H 2 → p1 + H, has an opportunity to take place. The second step of capping, reaction C 36 H 15 + C 2 H 2 → p1′ + H, which is slower than the first one to begin with, occurs more rarely as the fivemembered ring migration does not cease with the capping.
The elementary process underlying the migration of fivemembered rings over the aromatic edge is a "flip" reaction, 41 where the bond common to two adjacent rings switches from one ring to another, induced by the presence of a radical vacancy on the flipped-over ring. A reaction similar to edge five-membered rings can be envisioned for edge sevenmembered rings, as illustrated in Figure 7. The top panel of this figure demonstrates a "collision" of edge seven-and fivemembered rings forming two edge six-membered rings, and the bottom panel shows a migration step of an edge sevenmembered ring.
The flip reactions of edge seven-membered rings were included in the present kMC simulations. Considering the similarity in the underlying mechanism of the flip transformation, we assumed that the seven-membered-ring flip reactions occur with rates similar to those of five-membered rings. The latter were computed 39,41 to be on the order of 10 9 −10 10 s −1 , i.e., occurring much faster than other aromatic edge processes, and therefore could be modeled by partial The computed statistics for the flip reactions of sevenmembered ring are shown in Figure 8. As can be seen from the displayed histograms, the edge seven-membered rings clearly undergo migration, yet the length of the migration is rather moderate, just a few steps. The migration of seven-membered rings is rather rapidly terminated by them encountering fivemembered rings and thus forming two six-membered rings. This is also collaborated by the relatively short lifetime of seven-membered rings reported in the right panel of Figure 8.
The migration of five-membered rings and, as a consequence, the splitting of the fused five-membered rings of the E-bridge, along with the migration of the capping sevenmembered rings all lead rapidly to the "memory loss" of the initial E-bridge structure. This explains the computed closeness in the size of PAH structures computed with and without the E-bridge capping included in the kMC simulations and the similarity of their growth rate to that obtained with pyrene as the starting structure, shown in the middle panel of Figure 9. Also, the "collision" of migrating and chemisorbed five-and seven-membered rings forming six-membered rings during their encounters should lead to lower curvature. This, in fact, can be observed in the bottom panel of Figure 9, which displays a lower fraction of five-membered rings during the initial period of simulation, i.e., during the time period when such "collisions" and transformations occur.

CONCLUSIONS
At the conditions simulated in the present study, an Hmediated capping by C 2 H 2 of the fused five-membered-ring Ebridge occurred relatively frequently but not in all of the runs. It turned out that the embedded five-membered ring migration   The Journal of Physical Chemistry A pubs.acs.org/JPCA Article "split" the fused five-membered rings of the E-bridge rather quickly, before the capping takes place. Assuming similar elementary flip reactions occurring among edge seven-membered rings and edge five-and six-membered rings revealed rapid but short migration of the edge sevenmembered rings at their encounters with six-membered rings and the conversion of seven-and five-membered rings during their mutual encounters into a pair of six-membered rings.
The observed seven-membered ring flip and migration reactions along PAH edges obviously cannot be limited to only seven-membered rings formed in the capping considered in the present study but could happen to any edge seven-membered rings formed in other possible reactions. 42 While such reactions could be infrequent, 43 the forming seven-membered rings can be mutually "destroying" themselves and the encountered five-membered rings, by converting both into six-membered rings and thus reducing PAH curvature.